Finite quasisimple groups of 2×2 matrices over a division ring

1985 ◽  
Vol 97 (3) ◽  
pp. 415-420
Author(s):  
B. Hartley ◽  
M. A. Shahabi Shojaei
Keyword(s):  
Finite Groups ◽  
Division Ring ◽  
Characteristic Zero ◽  
Multiplicative Group ◽  
Main Tool ◽  
Number Fields ◽  
Soluble Groups ◽  
Trivial Element ◽  
Difficult Part

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

Subgroups of Central Separable Algebras

1973 ◽  
Vol 25 (4) ◽  
pp. 881-887 ◽  
Author(s):  
E. D. Elgethun
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Division Ring ◽  
Multiplicative Group ◽  
Prime Field ◽  
Odd Order ◽  
Multiplicative Subgroup ◽  
A Finite Group ◽  
Separable Algebras

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

Feldspar crystallization under magma-mixing conditions shown by cathodoluminescence and geochemical modelling – a case study from the Karkonosze pluton (SW Poland)

2004 ◽  
Vol 68 (4) ◽  
pp. 561-577 ◽  
Author(s):  
E. Słaby ◽  
J. Götze
Keyword(s):  
Magma Mixing ◽  
Main Tool ◽  
Precise Determination ◽  
Geochemical Modelling ◽  
Magma Evolution ◽  
Sw Poland ◽  
Crystallization Path ◽  
Geochemical Mass Balance

AbstractFeldspars from the Karkonosze pluton (SW Poland) display many features compatible with magma mixing. The mixing hypothesis has been tested using a geochemical mass balance law resulting in two possible paths of magma hybridization. Based on the results of the geochemical calculation, feldspar samples have been chosen along both mixing lines for cathodoluminescence (CL) investigation which was used as the main tool for the reconstruction of their crystallization path. Changes in the conditions of nucleation and crystallization of the feldspars as well as their movement within the magma chamber have been recognized due to different luminescence characteristics. These changes in the conditions of crystallization obtained by CL allow a precise determination of the genetic affinity of the samples to more mafic or more felsic environments.The results of the present study proved CL to be a valuable tool for the study of crystal-growth morphologies in a dynamic, turbulent environment and also as a geochemical tool for the reconstruction of various petrogenetic mechanisms (e.g. magma hybridization). Accordingly, the combination of CL with geochemical modelling provides corresponding information about magma evolution in an open system.

scholarly journals Quasinormal Fitting classes of finite groups

2019 ◽  
pp. 18-26
Author(s):  
Martsinkevich Anna V.
Keyword(s):  
Natural Number ◽  
Wreath Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Fitting Classes ◽  
Local Fitting ◽  
Class F

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If  X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially soluble groups.

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

Subnormal Subgroups of Division Rings

1963 ◽  
Vol 15 ◽  
pp. 80-83 ◽  
Author(s):  
I. N. Herstein ◽  
W. R. Scott
Keyword(s):  
Normal Subgroup ◽  
Division Ring ◽  
Multiplicative Group ◽  
Finite Sequence ◽  
Division Rings ◽  
Subnormal Subgroups

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

scholarly journals Schunck classes and projectors in a class of locally finite groups

1995 ◽  
Vol 38 (3) ◽  
pp. 511-522 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Finite Case ◽  
Saturated Formations ◽  
Definition Of ◽  
Schunck Class

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

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